Equation: Formula for Calculating a Skycraper’s Sway

Photo courtesy: Taipei 101

A skyscraper is a giant tuning fork. Give one a good knock — like with an earthquake or a heavy gust of wind — and it’ll start vibrating at its own natural resonance frequency (about seven octaves below the lowest notes on a piano). If you’re on the top floor of, say, the 1,667-foot-tall Taipei 101, you could find yourself swaying back and forth abruptly, a total of up to 2 feet within five seconds. Highly barfogenic. So Taipei 101’s designers hung a pendulum inside the building — in this case, they used an equation like the one below to determine that the megastructure needed a 730-ton weight with giant shock absorbers bolted to its bottom. It’s called a tuned mass damper, and when the tower starts to bend in the wind, the pendulum swings at the same frequency in the opposite direction, pulling the building upright and damping vibrations. It still sways, but subtly and smoothly. Here are the factors that come into play.

Damping (kg/s). This is the key variable. The goal is to halve acceleration, and more mass — building plus pendulum — means more damping.

Velocity of a swaying building at the top, in meters per second.

Acceleration (m/s2) at the top of a building — the main cause of nausea. It’s the derivative of velocity (), which is the derivative of distance (u).

The force applied to a structure by wind or an earthquake, measured in newtons (kg ⋅ m/s2).

Stiffness of a building, measured in newtons per meter (kg/s2) — the amount of force necessary to bend it a meter.

Displacement of a building at the top — that is, the distance it sways, in meters.

The dynamic mass of a building, in kilograms. The top of the structure will move more than the bottom, so the upper mass counts more.